Q. 94.1( 35 Votes )

# The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.

Answer :

Let a and r be the first term and the common ratio of the G.P. respectively.

∴ a_{1} = 1

nth term of G.P is given by -

a_{n} = ar^{n - 1}

a_{3} = ar^{2} = (1)r^{2}

a_{6} = ar^{4} = (1)r^{4}

According to question -

a_{3} + a_{6} = 90

⇒ r^{2} + r^{4} = 90

⇒ r^{4} + r^{2} – 90 = 0

Assume r^{2} = t

⇒ t^{2} + t – 90 = 0

⇒ t^{2} + 10t - 9t – 90 = 0

⇒ t(t + 10) - 9(t + 10) = 0

⇒ (t - 9)(t + 10) = 0

∴ t =9 or t = - 10(invalid because t = r^{2}, so t can't be - ve.)

∴ r^{2} = 9 ( Taking real roots)

∴ r = 3

Thus, the common ratio of the G.P. is 3.

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